"""This demo program solves Poisson's equation

    - div grad u(x, y) = f(x, y)

on the unit square with source f given by

    f(x, y) = 10*exp(-((x - 0.5)^2 + (y - 0.5)^2) / 0.02)

and boundary conditions given by

    u(x, y) = 0        for x = 0 or x = 1
du/dn(x, y) = sin(5*x) for y = 0 or y = 1
"""

from dolfin import *

# Create mesh and define function space
mesh = UnitInterval(32)
V = FunctionSpace(mesh, "CG", 1)

# Define boundary condition
gD = Constant(0.0)
class Kappa(Expression):
    def eval(self, value, x):
        if x[0] < DOLFIN_EPS:
            value[0] = 1.0e30
        else:
            value[0] = 0.0
kappa = Kappa()

# Define variational problem
v = TestFunction(V)
u = TrialFunction(V)
f = Expression("1.0")
gN = Expression("-1.0")
a = inner(grad(v), grad(u))*dx - kappa*u*v*ds
L = v*f*dx - kappa*gD*v*ds + v*gN*ds

U = Function(V)

# Assemble linear system
A = assemble(a)
b = assemble(L)
x = U.vector()

# Solve linear system
solver = LUSolver()
solver.solve(A, x, b)

# Save solution in VTK format
file = File("poisson.pvd")
file << U

# Plot solution
plot(U, interactive=True)